Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,14)(6,11)(7,10)(8,12)(9,13)(15,16)(17,18,22,25)(19,23,28,38)(20,30,32,40)(21,33,35,26)(24,34,39,31)(27,41,42,48)(29,45,44,37)(36,46,47,43);(1,3,8,4)(2,9,12,16)(5,11,14,10)(6,13,7,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(1,4,8,3)(2,5,12,14)(6,13,7,15)(9,11,16,10)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,38,47)(26,40,33,30)(29,46,44,43)(37,48,45,41);(1,5)(2,3)(4,12)(6,16)(7,9)(8,14)(10,13)(11,15);(1,6,8,7)(2,10,12,11)(3,13,4,15)(5,16,14,9)(18,25)(19,28)(24,39)(26,33)(27,42)(29,44)(30,40)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274848);